256 XIV INTEGRATION IN LOCALLY COMPACT GROUPS
virtue of (14.1.5) again, independent of the choice of left Haar measure //. The
mapping s\-+AG(s) is called the modulus function on G. If/is any /z-integrable
function, the function xt-*f(xs) is therefore also ^-integrable, and we have
f 1 f
(14.3.1.1) f(xs) d/x(*) = A(s" ') /(x)
J J
In particular, if A is any /i-integrable subset of G, then As is also ju-integrable,
and
(1 4.3.1 .2) n(As) = AfcXA).
(14.3.2) The mapping s\-+&G(s) is a continuous homomorphism ofG into the
multiplicative group R* .
This. follows immediately from the formula (14.3.1.1) and the lemma
(14.1.5.5).
The group G is said to be unimodular if AG(s) = 1 for all s e G. In this case
there is no distinction to be made between left and right Haar measures, and
we call them simply Haar measures on G.
(14.3.3) If there exists in G a compact neighborhood V of the neutral element e
which is invariant under all inner automorphisms of G, then G is unimodular.
This is the case, in particular, when G is compact, or discrete, or commutative,
For each SE G we have ji(V) = JJL^^VS) = A0(s()ju(V) by (14.3.1.2) and
(14.1.2.4); hence the result, since /u(V) ^ 0 (14.1.2).
(14.3.4) If u is any left Haar measure on G, then fi = A""1 • IJL. If f is any
H-integrable function on G, then the function x\-^f(x^l)^(x)~ 1 is u-integrable,
and we have
(14.3.4.1)
For each 5- e G we have
5C5XA-1 • /i) = (50A-1) • (8(^) = (A^^A-1) • (A^)^) = A"1 - u,
which shows that A""1 • ju is a right Haar measure on G. Since fi is also a right
Haar measure, there exists a constant a > 0 such that ji = aA""1 • /*. It follows
that n = ^(A"1 • //T = aA • ji — a2^ (13.14.5), whence a2 = 1 and thereforeoup